#120415

order	512
nilpotency class	3
isoclinism	isoclinic class of #120412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,10,7,12,5,11)(3,13,16,23,17,27,15,8)(4,18,19,26,20,25,21,9)(14,24,29,28,31,22,30,32)(33,34)(35,45)(36,41,51,58)(37,44)(38,42)(39,43)(40,47)(46,56,63,61)(48,59)(49,54,62,64)(50,60,53,57)(52,55);(1,3,5,15,7,17,6,16)(2,8,10,13,12,23,11,27)(4,14,19,29,20,31,21,30)(9,24,25,32,26,22,18,28)(33,35,37,47,39,48,38,52)(34,40,42,45,43,55,44,59)(36,49,50,46,51,62,53,63)(41,56,57,54,58,61,60,64);(1,4,5,19,7,20,6,21)(2,9,10,25,12,26,11,18)(3,14,15,29,17,31,16,30)(8,22,13,28,23,24,27,32)(33,36,37,50,39,51,38,53)(34,41,42,57,43,58,44,60)(35,46,47,62,48,63,52,49)(40,54,45,61,55,64,59,56)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27611);